Post-Newtonian reference ellipsoid for relativistic geodesy

Abstract

We apply general relativity to construct the post-Newtonian background manifold that serves as a reference spacetime in relativistic geodesy for conducting relativistic calculation of the geoid's undulation and the deflection of the plumb line from the vertical. We chose an axisymmetric ellipsoidal body made up of perfect homogeneous fluid uniformly rotating around a fixed axis, as a source generating the reference geometry of the background manifold through Einstein's equations. We, then, reformulate and extend hydrodynamic calculations of rotating fluids done by a number of previous researchers for astrophysical applications to the realm of relativistic geodesy to set up algebraic equations defining the shape of the post-Newtonian reference ellipsoid. To complete this task, we explicitly perform all integrals characterizing gravitational field potentials inside the fluid body and represent them in terms of the elementary functions depending on the eccentricity of the ellipsoid. We fully explore the coordinate (gauge) freedom of the equations describing the post-Newtonian ellipsoid and demonstrate that the fractional deviation of the post-Newtonian level surface from the Maclaurin ellipsoid can be made much smaller than the previously anticipated estimate based on the astrophysical application of the coordinate gauge advocated by Bardeen and Chandrasekhar. We also derive the gauge-invariant relations of the post-Newtonian mass and the constant angular velocity of the rotating fluid with the parameters characterizing the shape of the post-Newtonian ellipsoid including its eccentricity, a semiminor and a semimajor axes. We formulate the post-Newtonian theorems of Pizzetti and Clairaut that are used in geodesy to connect the geometric parameters of the reference ellipsoid to the physically measurable force of gravity at the pole and equator of the ellipsoid. Finally, we expand the post-Newtonian geodetic equations describing the post-Newtonian ellipsoid to the Taylor series with respect to the eccentricity of the ellipsoid and discuss their practical applications for geodetic constants and relations adopted in fundamental astronomy.

Reference Ellipsoid and Geoid in Chronometric Geodesy

“…It describes the figure of a fluid body with a homogeneous mass density rotating around a fixed z-axis with a constant angular velocity ω. The post-Newtonian effects deforms the shape of Maclaurin's ellipsoid (Chandrasekhar, 1965;Bardeen, 1971) and modify the basic equations of classic geodesy (Müller et al, 2008;Kopeikin et al, 2011Kopeikin et al, , 2016. These relativistic effects must be properly calculated to ensure the adequacy of the geodetic coordinate transformations at the millimeter level of accuracy.…”

Theoretical Tools for Relativistic Gravimetry, Gradiometry and Chronometric Geodesy and Application to a Parameterized Post-Newtonian Metric

“…Recently, extensive work has been done aiming at developing an exact relativistic theory of Earth's geoid undulation [26], as well as developing a theory of the reference level surface in the context of post-Newtonian gravity [27,28]. This goes beyond the problem of the realization of a reference isochronometric surface and tackles the tough work of extending all concepts of classical physical geodesy (see, e.g., [23]) in the framework of general relativity.…”

“…This formula was derived for the metric (27) to (29) with w = 0 and general functions W( x) and W( x) (U * (x i ) and V j (x i ) in the notation of [32]) and using the tetrad (35) to (38). The terms with w can be added by substitution of a corresponding function for V j .…”

“…We find a tetrad that satisfies e α 0 = u α /c and Equation (8) for the metric (27) to (29), valid for a general trajectory with coordinate velocity v i :…”

“…It is less than 3.6 × 10 −14 for a satellite and around 2.2 × 10 −15 on the ground. In [51], the term (24) is calculated for the metric given later in Equations (27) to (29) up to the fourth order. It is stressed that the J 2 term of the expansion (30) in the third order term can amount to 1.3 × 10 −16 for a satellite in low orbit.…”